Having trouble finding lenght

[Zulgok]

Hi, I have to find AO₂ lenght. What I was given is A₁O₁=3, AO₁=6. Space between two circles could be AO₂ middle, but I have nothing to prove it, because I dont know anything about that larger circle, only that its proportional. What is there that I dont notice ? If possible, Id love to receive a direction or hint .

 

[pka]

Hi, I have to find AO₂ lenght. What I was given is A₁O₁=3, AO₁=6. Space between two circles could be AO₂ middle, but I have nothing to prove it, because I dont know anything about that larger circle, only that its proportional. What is there that I dont notice ? If possible, Id love to receive a direction or hint .

\(\displaystyle \Delta AA_1O_1~\&~\Delta AA_2O_2\) are similar right triangles. Let \(\displaystyle D\) denote the point at which the circles are tangent. Then \(\displaystyle A_1O_1=O_1D~\&~DO_2=O_2A_2\) .
Therefore use proportions.

 

[Zulgok]

Out of [FONT=MathJax_Math-italic]A[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math-italic]O[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]O[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math-italic]D[/FONT][FONT=MathJax_Main] [/FONT][FONT=MathJax_Main]&[/FONT][FONT=MathJax_Main] [/FONT][FONT=MathJax_Math-italic]D[/FONT][FONT=MathJax_Math-italic]O[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]O[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math-italic]A[/FONT][FONT=MathJax_Main]2[/FONT]A1O1=O1D & DO2=O2A2 proportion I got that AO₂ is 3 times bigger than AO₁. So AO₂​=18. Is this correct ?